Abstract: We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate for the number of labelled planar graphs on vertices, where and are explicit computable constants. We show that the number of edges in random planar graphs is asymptotically normal with linear mean and variance and, as a consequence, the number of edges is sharply concentrated around its expected value. Moreover we prove an estimate for the number of planar graphs with vertices and edges, where is an analytic function of .

We also show that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law , where is an explicit constant.